Advanced Flow Simulation in Aricraft Design

8/12/2019 Advanced Flow Simulation in Aricraft Design

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Advanced flowsimulation methods inaircraft design

Mattias Sillen

The author

Mattias Sillen is at Saab Aerospace, SE-581 88 Linkoping,

Sweden

Keywords

Aircraft, Aerodynamics, Computational fluid dynamics,

Turbulence

Abstract

The compressible Navier-Stokes equations are solved

numerically for turbulent transonic aerospace applications

on parallel computers. An Explicit Algebraic Reynolds

Stress Model (EARSM) models the turbulence. Expressing

the EARSM as an extension of an eddy-viscosity model

makes the implementation straightforward in a flow solver

with existing two-equation eddy-viscosity models. The

k2 vtransport equations are used as a platform for the

model. The EARSM approach significantly improves the

shock position for transonic flow over wings without

substantial increase in computational cost. Industrial use

of advanced flow modelling requires a short turn-around

time of computations. This is enabled through the use of

parallel computers. To achieve good parallel performance

the computational load has to be evenly distributed

between the processors of the parallel computer. A

heuristic algorithm is described for distributing and

splitting the blocks of a structured multiblock grid for agood static load balance. Speed-up results are presented

for turbulent flow around a wing on a number of parallel

platforms.

Electronic access

The current issue and full text archive of this journal is

available at

http://www/emeraldinsight.com/0002-2667.htm

Introduction

In the aerospace industry today

Computational Fluid Dynamics (CFD) plays

an increasingly important role as a tool for

design and analysis. Typical characteristics for

the industrial CFD environment are the often

complicated geometry, the frequency of large

problems, the requirement of high fidelity

results, the short time schedule for producing

results, etc. A variety of methods are used

from preliminary design studies where short

problem turn-around time is crucial to

detailed component analysis where accurate

results are required. To further improve the

confidence of the predictions in the

aerodynamic design process, advanced CFD

methods, i.e. Reynolds Average Navier-Stokes (RANS) based 3D methods, must be

deployed at an even earlier stage than today.

Early in the design process when the geometry

is rapidly changing, the allowable time frame

to produce a complete flow analysis is no more

than a day. To be able to meet this goal the

CFD program execution time has to be less

then 15 hours, i.e. overnight. This allows for

evaluation of the results and preparation of an

improved design during the working day. A

cost-efficient way to reduce the problem turn-around time is to use parallel computers and

efficient numerical algorithms to solve the

flow equations. To achieve good parallel

performance of a CFD code the

computational load has to be evenly

distributed between the processors of a

parallel computer. An algorithm is here

described how to partition structured

multiblock grids for efficient use of parallel

computers.

Until recently the prevailing industrialstate-

of-the-art CFD methods (Gould et al., 2000)

included viscous coupled Euler solvers or

RANS-methods with algebraic turbulence

closures. To be able to deal with complex

flows and obtain results with a higher degree

of confidence more advanced methods are

required. Methods based on RANS equations

with two-equation turbulence models or

higher have been around in the research

community for a long time but just recently

entered the industrial aerodynamic design

process. Explicit Algebraic Reynolds StressModels (EARSM) has emerged as an

industrially feasible class of turbulence

models offering improved physical modelling

compared to linear eddy-viscosity models

Aircraft Engineering and Aerospace Technology

Volume 74 Number 2 2002 pp. 138146

q MCB UP Limited ISSN 0002-2667

DOI 10.1108/00022660210420843

138
http://www.emeraldinsight.com/0002-2667.htmhttp://www.emeraldinsight.com/0002-2667.htmhttp://www.emeraldinsight.com/0002-2667.htm


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without being as computationally expansive as

the more complete Reynolds stress models.

The present article describes recent

development in turbulence modelling and

parallel implementation in one of the RANS

methods used for aerodynamic design and

analysis at Saab Aerospace. In the next section

are the turbulence models described with an

emphasis on the EARSM approach. The

numerical implementation is discussed in the

subsequent section. This is followed by a

description of the parallel load balancing

strategy. Numerical results showing the

benefit of using EARSM compared to a

standard two-equation eddy-viscosity model

for a number of transonic wings conclude the

paper, where also speed-up results are

presented for different parallel platforms.

Turbulence modelling

The compressible Navier-Stokes equations

model the flow of air around aircraft very well.

For such applications the influence of

turbulence must be modelled, because of

insufficient memory capacity and

computational speed of present computers to

solve the turbulent flow equations directly.

From an industrial point of view, a suitable

compromise between robustness, efficiency

and validity for turbulence closure has often

been found in the class of two-equation eddy-

viscosity models represented by, e.g. thek 2 e

or thek 2 vmodels. A frequently used model

in the aerospace industry is the Wilcoxk 2 v

model (Wilcox, 1993). Primarily due to its

numerical robustness and that it does not

need any wall-distance information, which

can be cumbersome to compute for complex

three-dimensional configurations. In the

Wilcox standard k 2 vmodel the transport

equations read:

D

Dtrk P2 r1

xjmmt

sk

k

xj

1

D

Dtrv a v

kP2 brv2

xj mm t

sv v

xj 2

where k is the turbulent kinetic energy, vis the

specific dissipation rate ande ; b*vkis the

dissipation rate. The coefficients a5=9;

b3=40; b* 9=100;sk2:0 andsv2:0 are the model constants.

The production of turbulent kinetic energy,

P, is defined as:

P ;2ruiujUi

xj 3

In a two-equation eddy-viscosity model the

Reynolds stresses are modelled as

ruiuj23rkdij 2 2mtS

*ij 4

where

S*ij 1

2

Ui

xj Uj

xi2

2

3

Uk

xkdij

5

and the eddy-viscosity is defined as

mtrkv

6

One often noticed drawback of this model is

its free-stream dependency. A recent proposal

(Kok, 2000) resolves this effectively by

introducing a diffusion correction that has

negligible additional cost and is free of

inconvenient parameters such as local wall-

distances. A cross-diffusion term is added to

thevtransport equation, Equation (2).

CDsd rv

max k

xi

v

xi; 0

7

sd0:5 is an additional model constant. Thevalue of the model parameter skis here

modified to 1.5.

For more demanding flow cases where flow

separation or streamline curvature is present

the standard two-equation models often fail to

produce accurate results. This is often due to

the linear stress/strain relationship. A

promising way to introduce enhanced physics

with a computational cost comparable to the

existing two-equation models is to use the

non-linear stress/strain relationship offered by

an EARSM. The EARSM proposed by Wallin

and Johansson (2000) is derived from a

differential Reynolds Stress Model by

Launder et al. (1975). A self-consistent and

fully explicit algebraic relation for the

Reynolds stresses in terms of the mean flow

field is obtained by applying the so-called

equilibrium assumption, where theadvection and diffusion of the Reynolds stress

anisotropy are neglected. The model is based

on thek2vtransport equations as given in

Equations (1) and (2).

Advanced flow simulation methods in aircraft design

Mattias Sillen


Volume 74 Number 2 2002 138146

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In the EARSM the Reynolds stresses may

be written in terms of an effective turbulent

viscositymteffand an extra anisotropy aexij :

ruiuj 23 rkdij2 2mteffS*ij rkaexij 8

where the effective turbulent viscosity is

mteff 2 12b1IIVb6rkt 9

and the extra anisotropy becomes

aexb3 V2 2 13 IIVI b4SV2VS

b6 SV2 V2S2 IIVS2 23 IVI

b9VSV2 2V2SV10

The invariants are defined by

IIStr{S2}; IIVtr{V2};

IVtr{SV2}11

Herea,SandVdenotes second rank tensors,

tr{ } denotes the trace and I is the identity

matrix. The inner product of two matrices is

defined asSSij ; S2ij ; SikSkj: Thenormalised mean strain- and rotation rate

tensors are defined as

Sij tS*ij;VijtV*ij 12

where the turbulent time-scale is defined by

tmax k1

; Ct

ffiffiffiffiffim

r1

r 13

andCt6:0 is a model constant.The dimensional strain- and rotation rate

tensors are

S*ij

1

2

Ui

xj

Uj

xi

22

3

Uk

xk

dij and V*ij

12

Ui

xj2

Uj

xi

14

Theb-coefficients are given by

b1 2N2N22 7IIV

Q ;

b3 212N21IV

Q ;

b4 22N22 2IIVQ

;

b6 26NQ

; b9 6Q

15

with the non-singular denominator

Q56N2 2 2IIV2N2 2 IIV 16

N is given by

N

c013 P1 ffiffiffiffiffiffiP2p 1=3

signP1 2ffiffiffiffiffiffi

P2p jP1 2

ffiffiffiffiffiffiP2

p j1=3;P2 $ 0

c013 2P21 2 P21=6

cos 13

arccos P1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P21 2 P2

q0B@

1CA

0B@

1CA;

P2 , 0

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: 17

with

P1 c021

27 9

20IIS2

2

3IIV

c01;

P2P21 2 c

021

9 9

10IIS2

3IIV

3 18

and

c01 94c1 2 1 19

c11:8 is a model constant.

Numerical implementation

The Navier-Stokes equations are a non-linear

system of time-dependent partial differential

equations to solve for the density r, the

momentum components in the three

Cartesian co-ordinate directions rui; i1; 2; 3; and the total internal energy rEin the

field. The flow equations are approximated on

a structured grid with a finite volume

approximation using central differencing. A

blend of second and fourth order artificial

viscosity is added to avoid oscillatory

behaviour in the smooth parts and in the

vicinity of shocks (Jameson et al., 1981,

Jameson, 1988). The flow equations are

integrated forward in time by a Runge-Kutta

method until the time derivatives are

sufficiently small and we are close to a steady

state solution. The convergence to steadystate is accelerated by a multi-grid method,

where the solutions on a sequence of coarser

grids are combined to improve the

convergence rate.


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The numerical implementation of thek2v

transport equations is implicit, solving the

steady transport equations by ADI technique.

One relaxation sweep is performed in each

direction after every complete Runge-Kutta

cycle for the flow equations. The spatialderivatives are evaluated using the same finite

volume technique as the flow equations but

with a hybrid central/upwind differencing.

The Reynolds stresses expressed by the

EARSM are written in terms of an effective

turbulent viscosity mteffand an extra

anisotropyaexij : The reason for that is that

most flow solvers with two-equation eddy-

viscosity models can reuse subroutines when

the model is formulated using an effective

turbulent viscosity. It has shown that it is

sufficient to only consider the effective

turbulent viscosity when determining the

stability limit for the flow equations. The extra

terms associated with the extra anisotropy can

be treated as a fully decoupled source term

(Wallin, 1999).

The production of the turbulent kinetic

energy in a two-equation eddy-viscosity

model, P(EVM), is expressed by using the

assumption in Equation (4).

PEVM 2mtS*ij 2 23 rkdij

Uixj

20

Now, using the effective turbulent viscosity

defined in (9) in relation (4) and by adding the

contribution from the extra anisotropy the

production of the turbulent kinetic energy in

the EARSM is written as

PPEVM 2 rkaexijUi

xj21

This term is the most important to modify toinclude effects of streamline curvature and

local and global rotation. The second most

important term to modify is the turbulent

transport of momentum in the momentum

equation. In a two-equation eddy-viscosity

model the term is modelled as

TEVMi

xj2mtS

*ij 2

23rkdij

22

In the EARSM a more physical sound

modelling of this term is

TiTEVMi 2

xjrkaexij 23

using the effective turbulent viscosity in (9).

An alternative to EARSM based on

standardk 2 v; that is even simpler to

implement, is the so-called linear EARSM. It

is actually an eddy-viscosity model where only

the eddy-viscosity part of relation (8) is kept.

This means thataexij

0:Some of the physics

covered of the full EARSM is naturally lost in

this approach. It is however very easily

introduced into an existing two-equation

eddy-viscosity model where only the routine

that computes the eddy-viscosity needs to be

modified. Compared to a standard eddy-

viscosity model with a constant Cmit is based

on a sounder basis.

The same physical boundary conditions are

used fork andvin the EARSM as in the two-

equation implementation. The physical

conditions are

k0 and v! 6mbry2

24

For the applications so far the EARSM has

showed the same numerical behaviour as the

standard Wilcoxk2vand the computational

cost is approximately 8 per cent higher than

for the standard Wilcox.

Parallel implementation and loadbalancing

The code described here was originally

developed for the Cray vector-computers.

The effort to parallelize it was modest because

of the multiblock structure already introduced

in the serial version. A master-slave model is

used where a master process initialises and

supervises the iterations. Each slave process is

responsible for a subset of the blocks. The

communication is implemented using PVM

for portability between different platforms.

The difference stencil at a block boundary

needs solution data from the neighbouring

block on the other side of the boundary. These

data are sent over the network if the

neighbouring block is located on another

processor or moved internally in the memory

if the processor also take care of the

neighbour. Updated boundary data are

transferred between neighbours at every stage

or at the end of a full Runge-Kutta step. This

is also the case when multiple grids are used.Of particular importance in parallel

computing is the efficiency. Multi-grid

acceleration can improve the convergence rate

and lower the number of iterations


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substantially in both serial and parallel mode.

In our parallel implementation the same

processor handles the same blocks on all grid

levels. In this way, all processors are kept busy

also on the coarsest grid, even if the quotient

between useful computing time andcommunication time decreases. The load

balance is critical to good performance. A

usual assumption is that the grid can be

partitioned arbitrarily. This is not the case

here, as we have a block structure from the

beginning created by a grid generator system.

It is assumed that the original partitioning of

the grid into blocks is determined only by the

topology of the configuration. The blocks are

distributed based on an analysis of the

computational and communication time

before the iterations start for optimal use of

the computer. If an even load is not obtained

with the original blocks, then they are split

until a convergence criterion is satisfied.

The CPU time for a time-step on a

processor is proportional to the number of

grid cells in the blocks on the processor. The

time of transferring block boundary data from

one processor to another depends on the

number of boundary cells the blocks on the

processors have in common. An estimate of

the total time tfor a time-step with a given

partition ofNblockblocks onNprocprocessors,

C, is

tC k1;Nproc

maxtCPUk XNprocn1

tcommn 25

wheretCPUk is an estimate of the CPU-time for

a time-step for the blocks on the processor k.

The CPU-time is based on the number of

floating-point operations per cell and the

processor speed. Accurate estimates of thenumber of floating point operations per cell

are available for different equations. The

communication timetcommk for sending data

from one processork to all other processors is

based on the number of cells the processors

have in common, the number of variables to

communicate and the bandwidth of the

shared network, see (Alund et al., 1997) for a

more detailed analysis.

The algorithm tries to find the distribution

Cwith the minimum total execution timet(C). TheNblockblocks in the grid are

distributed into Nproc subsets of blocks in a

pre-processing step. The subsets are

determined statically.

(1) Sort the blocks in decreasing order

according to the execution timeti; i1;. . .;Nblock assuming that the problem is

solved on a parallel computer with Nblockidentical processors processing one block

on each processor.

(2) Start with empty subsets.

(3) Then each block is taken from the sorted

set and inserted into the subset that gives

the smallest value oftfor the partitioning

built up so far.

Sometimes it is not possible to obtain a good

balance with the original sizes of the blocks.

Then it is necessary to split some of the blocks

and re-compute the load distribution to see if

a more satisfactory solution has been found.

To keep the quotient between computation

time and communication time large, we

should split as few blocks as possible. When a

block has been selected for splitting it is

divided in the middle of the longest edge, so

the two new blocks introduce as little extra

communication as possible.

There is no guarantee that this heuristic

algorithm above finds the optimal solution. It

will however produce an acceptable load

balance even though based on a simple

communication model with the advantage of

being computationally inexpensive.

Numerical examples

In this section are a number of industrially

relevant examples of viscous compressible

flow problems presented where the benefit of

using advanced turbulence models is clearly

demonstrated. The examples cover a 2D

airfoil, a 3D wing and a complete Airbus

configuration in transonic flow. The

advantage of using parallel computers to

achieve a short problem turn-around time is

exemplified with viscous flow around the

ONERA M6 wing.

RAE 2822 Airfoil

The two-dimensional airfoil RAE 2822 in

transonic flow conditions is a typical example

of a demanding aerospace application. The

flow condition is referred to as Case 10, i.e.

M1

0:754;a

2:578 andRec

6:2 106:

The airfoil is highly loaded and a boundarylayer separation occurs at the shock

impingement point. The experimental data

are from Cocket al., 1979. The

computational mesh consists of 272 cells in


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the stream-wise direction and 80 in the

normal direction see Figure 1. Transition is

prescribed at 3 per cent of the chord.

In Figure 2 are the pressure coefficient

distributions presented for computations

using different turbulence models. The

applied models include Wilcoxk2v, k 2 v

with cross-diffusion according to Kok,

EARSM based on Wilcox, EARSM based on

Kok and linear EARSM based on Wilcox.

The different EARSM perform significantly

better than the linear eddy-viscosityk 2 v

models even though there is still a small

mismatch in shock location compared to the

measurements. The results produced by the

linear EARSM fall in between the full

EARSM and thek 2 vresults. There are

other computations published on this casethat have better agreements. The reason for

this is that there exists a number of different

geometries for this case, the measured or

design geometry with or without an additional

camber correction. In this case the measured

geometry is used with the camber correction.

In Figures 3 and 4 are the velocity profiles

computed with the different turbulence

models compared to measurements. At

locationx=c0:404; i.e. upstream of theshock, the different models predict almost the

same velocity profile. Downstream of the

shock, at x=c0:900;the scatter is greaterbut the more advanced models compare

better with measurements.

LANN wing

The LANN wing is a supercritical transport

wing, which has been measured at transonic

wind tunnel conditions (Horstenet al., 1983).

The computational mesh consists of 192 cells

in the stream-wise direction, 48 cells in the

span-wise direction and 64 in the normal

direction. The wing surface is discretised by

128 32 cells. The on-flow conditions are

M10:82;a2:608 andRemac5:32 106:

The flow is characterised by a strong shock

followed by a boundary layer separation on

the upper surface, see computational results

in Figure 5.

The EARSM predicts a larger separatedflow region compared to thek2vmodel. The

computed pressure distribution is

dramatically improved by using the EARSM

instead of the k 2 vmodel as seen in Figures 6

and 7. The computed pressure distributions

are compared with measurements at 47.5 per

cent and 82.5 per cent of the span. The

Figure 1Computational grid around RAE 2822 airfoil

Figure 2Cpdistribution computed with different turbulence models on RAE

2822 compared with measurements. Flow case M10.754,a2.578 andRec6.2 106.

Figure 3Velocity profile atx/c0.404 computed with different turbulencemodels compared with measurements. Flow caseM

10.754,a2.578andRec

6.2 106.


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EARSM based on the k 2 vmodel with the

cross diffusion term performs very well,

predicting the same shock location as in the

measurements.

Airbus-like configuration

The Airbus-like AS28 wing-body-pylon-

nacelle configuration is a demanding example

with respect to the geometrical complexity

and computational mesh size. It is a typical

example of an industrial aerospace

application.

The AS28 configuration has been wind

tunnel tested by Aerospatiale in different

configurations. A structured multiblock grid

is generated with a total of 270 blocks and

almost 5 million cells. The considered flow

case is defined by M10:80; a2:28 andRemac

9:97 106:The nacelle has through-

flow conditions. The surface pressure

distribution as given by the EARSM is

presented in Figure 8.

Computational results obtained with

standard Wilcoxk2vare compared with

results produced by EARSM based on k 2 v

with cross-diffusion according to Kok and

wind tunnel measurements at 17 per cent and

57 per cent of the span, see Figure 9 and 10.

Figure 4Velocity profile at x/c0.900 computed with different turbulencemodels compared with measurements. Flow caseM10.754,a2.578andRec6.2 106.

Figure 5Surface pressure distributions and skin friction

lines on upper side of the LANN wing as computed by the

Wilcoxk2 vmodel (upper) and by the EARSM based on

k2 vwith cross-diffusion by Kok (lower). The on-flow

conditions areM10.82,a2.608 and

Remac5.32 106.

Figure 6Pressure distribution at y/b47.5 per cent as computed withdifferent turbulence models compared with measurements

Figure 7Pressure distribution at y/b82.5 per cent as computed withdifferent turbulence models compared with measurements


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The pressure distributions predicted by the

different turbulence models compare very

well on the lower part of the wing. On the

upper side the shock position is more

accurately predicted by the EARSM while thestandard k 2 vmodel predict a too far down-

stream position. The pressure at the trailing

edge is also better predicted by the EARSM.

Both models have minor problems predicting

the right pressure level just down-stream of

the shock. The flow over the wing is close to

separation at the shock and at the trailing

edge.

The typical run time for an application of

this size is around 40 h on a Cray T3E using

30 processors.

Parallel performance

The problem turn around time is of

importance when using advanced flow

modelling at an early stage in the design cycle.

Here is the parallel efficiency of the flow solver

evaluated on different parallel platforms using

computation of the turbulent flow field

around the ONERA M6 wing at free-stream

conditionsM10:84; a3:068 andRemac11:7 106: The surface pressure onthe wing is presented in Figure 11. The

computational mesh consists of 1.2 million

cells divided in 32 blocks. The flow is

modelled by the Wilcox k2vmodel. Load

balancing of the computation is performed

with the algorithm described in previous

section. The tested computers include Cray

T3E, SGI 3000 and a Linux PC-cluster. Ascan be seen in Figure 12 the speed-up is nearly

linear for the evaluated parallel computers.

The Cray T3E scales very well up to 128

processors (not fully included in the figure),

where a speed-up of 116 is achieved. The

latest tested computer, a Linux PC-cluster

consisting of Pentium III processors

connected via a Fast Ethernet network,

performs well up to 16 processors. Only a

slight degradation of the speed-up is seen for

16 processors, where the network is starting to

Figure 8Surface pressure predicted by the EARSM at

M10.80,a2.28 andRemac9.97 106.

Figure 9Pressure distribution at y/b17 per cent. Computations withEARSM andk2 vmodels compared with measurements

Figure 10Pressure distribution at y/b57 per cent. Computations withEARSM andk2 vmodels compared with measurements

Figure 11Surface pressure distribution on upper side of

ONERA M6 at free-stream conditions M10.84,

a3.068 andRemac11.7 106.


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become saturated. The SGI 3000 shows a

similar behaviour as the Linux-cluster, the

curves fall on each other in the figure. For 16

and 32 processors the speed-up is slightly

deteriorated. The excellent parallel

performance of the Cray T3E compared to

the SGI 3000 and the PC-cluster is partly

explained by the slower processor type in the

T3E and partly by the high performing

internal network. When comparing MFloprates the SGI 3000 outperforms the other

computers.

This example shows that this type of

explicit CFD code performs very well on

parallel computers. To achieve a balanced

load when using many processors the original

32 blocks are subdivided as described in the

previous section. The results show that the

load-balancing algorithm works well,

introducing only a minor loss in efficiency

compared to the theoretical speed-up.

Conclusions

The physically more advanced modelling of

turbulent flow, as given by an EARSM, is

shown to give clearly improved results for a

number of transonic aerospace applications

compared to the traditional two-equation

eddy-viscosity models represented by the

Wilcox k 2 vmodel.

For a flow simulation system to be

applicable in the early stages of the design

process, the problem turn-around time must

be kept short. The use of parallel computers is

a way to significantly reduce the CFD-

program execution time. An algorithm is

described how to partition a structured multi-

block grid for efficient use on parallel

computer platforms. Good speed-up numbers

are reported for various parallel computers on

a wing case.

References

Alund, A., Lotstedt, P. and Sillen, M. (1997), Parallel

single and multigrid solution of industrial

compressible flow problems,Computers & Fluids,

26 No. 7, pp. 775-91.Cock P. H., MacDonald M. A. and Firmin M. C. P. (1979)

Aerofoil 2822 Pressure distribution, boundary

layer and wake measurements, AGARD AR 138.Gould A., Courty J-C., Sillen M., Elsholz E. and Abbas A.

(2000) The AVTAC Project a Review of European

Aerospace CFD, Proceedings ECCOMAS 2000.Horsten J. J., Den Boer R. G. and Zwaan R. J. (1983)

Unsteady transonic pressure measurements on a

semi-span wind tunnel model of a transport-type

supercritical wing (LANN model), AFWAL-TR-83-

3039.Jameson A. (1988) Computational transonics,

Communications in Pure and Applied Mathematics,

XLI, pp. 507-49.Jameson A., Schmitt W. and Turkel E. (1981) Numerical

solution of the Euler equations by finite volume

methods using Runge-Kutta time-stepping schems,

AIAA paper 81-1259.Kok, J. (2000), Resolving the dependency on free-stream

values for the k2 vturbulence model, AIAA

Journal, 38 No. 7.Launder, B.E., Reece, G.J. and Rodi, W. (1975), Progress in

the development of a Reynolds-stress turbulenceclosure,J. Fluid Mech., 41, pp. 537-66.

Wallin S. (1999) An efficient explicit algebraic Reynolds

stress k2 vmodel (EARSM) for aeronautical

applications, FFA TN 1999-71.Wallin, S. and Johansson, A.V. (2000), An explicit

algebraic Reynolds stress model for incompressible

and compressible turbulent flows, J. Fluid Mech.,

403, pp. 89-132.Wilcox D. C. (1993) Turbulence modeling for CFD, DCW

Industries Inc., La Canada, California.

Figure 12Speed-up numbers versus number of processors using different

parallel computer platforms


Mattias Sillen



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Advanced Flow Simulation in Aricraft Design